It is given that BF and AB are congruent. Let's indicate this on the given diagram.
We are asked to show that ABFH is a rhombus. Since two of the consecutive sides are congruent, it is enough to show that ABFH is a parallelogram. It is also given that ACDH and BCDF are parallelograms. First, let's focus on parallelogram ACDH.
Opposite sides of parallelograms are parallel and, according to Theorem 6.3, they are also congruent.
AH∥CD
AH≅CD
Quadrilateral BCDF is also a parallelogram.
The opposite sides of this parallelogram are also parallel and congruent.
CD∥BF
CD≅BF
We can combine our results.
AH∥CD∥BF
AH≅CD≅BF
According to the Transitive Property, segments AH and BF are parallel and congruent. These segments are opposite sides of quadrilateral ABFH. According to Theorem 6.12, this guarantees that ABFH is a parallelogram.
Since it is given that consecutive sides BF and AB are congruent, we can use Theorem 6.19 to conclude that this parallelogram is a rhombus. We can summarize these steps in a two-column proof.
Completed Proof
2
&Given:&& ACDH is a parallelogram
& && BCDF is a parallelogram
& && BF≅AB
&Prove:&& ABFH is a rhombus
Proof:
Statements
Reasons
1.
ACDH is a parallelogram
1.
Given
2.
AH∥CD
2.
Definition
3.
AH≅CD
3.
Opposite sides of a parallelogram (Theorem 6.3)
4.
BCDF is a parallelogram
4.
Given
5.
CD∥BF
5.
Definition
6.
CD≅BF
6.
Opposite sides of a parallelogram (Theorem 6.3)
7.
AH∥BF, AH≅BF
7.
Transitive property
8.
ABFH is a parallelogram
8.
Opposite sides are parallel and congruent (Theorem 6.12).
